IIR filter design process starts with reference analog prototype filter. This book explains Butterworth, Chebyshev (Chebyshev I) and inverse Chebyshev filter (Chebyshev II).

The transform function of analog filter Hsa(s) is expressed as:
where:
N is the filter order;
s is the complex frequency (s = σ + jΩ); and
M ≤ N.

In order that a system described via expression above is stable, it is necessary that all poles (the square roots of polinomial Aa(s)) are located in the left half of S plane. Figure 3-3-1 illustrates S plane.  
A low-pass filter is used for analog filter design. The conversion into the appropriate type of filter (high-pass, band-pass or band-stop) is performed by converting into analog filter, i.e. frequency axis scaling.

3.3.1 Butterworth analog filter

Low-pass Butterworth analog filters are filters whose frequency response is a monotonious descending function. They are also known as „maximally flat magnitude“ filters at the frequency of Ω = 0, as the first 2N - 1 derivatives of the transfer function when Ω = 0 are equal to zero.

Butterworth filter is characterized by 3dB attenuation at the frequency of Ω=1, no matter the filter order is. Figure 3-3-2 illustrates frequency responses for a few various parameters N (filter order).  
Butterworth filter is defined via expression:
where:
Ω is the frequency; and
N is the filter order.  
Figure 3-3-3 illustrates IIR filter specification with parameters of most interest for Butterworth filter.

To design Butterworth reference analog prototype filter, it is necessary to know the filter order. All poles of the resulting filter must be located in the left half of the S plane, i.e. to the left of the imaginary axis.

Note:

In the z plane, filter is stable if all poles are located within the unit circle. In the s plane, filter is stable if all poles are located in the left half of the s plane. Z-transform is used for digital filters, whereas Laplase transform (s plane) is used for analog filters. Even though these two types of transformations are similar to some extent, they should not be mixed up concerning the filter stability analyse.

When the filter order is known, it is easy to find its poles using expression:
Butterworth poles are equally allocated (equidistantly) on the unit circle within the left half of the s plane. The location of poles for N=5 and N=6 is shown in Figure 3-3-4.  
The transfer function of the Butterworth reference analog prototype filter is expressed as follows:
where:
Sk is the k-th pole of the Butterworth filter transfer function
For N=5, the transfer function is:  

3.3.2 Chebyshev Analog Filter

Chebyshev analog low-pass filter of the first kind is a type of analog filter that has the least oscillation in frequency response in the entire passband. Therefore it is characterized by equal ripple in the passband and the stopband frequency response is monotoniously descending function.

Figure 3-3-5 illustrates frequency response for a 4th order band-stop Chebyshev reference analog filter.  
To design Chebyshev reference analog prototype filter, it is necessary to know the filter order.

Chebyshev analog filter is defined via expression:
where:
Ω is the frequency;
N is the filter order;
ε is a parameter used to define maximum oscillations in the passband frequency response; and
TN is the Chebyshev polynomial.

The Chebyshev polynomial TN(Ω) can be obtained via recursive relations:
If the filter order is known in advance, neither recursive relations nor expression for the square of frequency response are necessary. The design process starts from the values of poles of a 1st order Chebyshev reference analog filter.

The values of poles are expressed as:
where:
si is the i-th transfer function pole of analog prototype filter (complex value);
σi is the pole; and
Ωi is the imaginary pole.
where:
N is the filter order; and
i=1, 2, ..., N.

The value of parameter ε is obtained via expression:
Transfer function is expressed as:
The value of A0 is found via expression:
For N=5, the transfer function is:      

3.3.3 Inverse Chebyshev Analog Filter

Inverse Chebyshev analog filter is also known as Chebyshev analog filter of the second kind. The frequency response of this filter monotoniously falls in the passband and transition region. Similar to Butterworth filter, the frequency response is extremely flat function at the frequency of Ω = 0, as the first 2N - 1 derivatives of the transfer function for Ω = 0 are equal to zero. In the stopband, inverse Chebyshev filter has the least oscillation in the frequency response.

Figure 3-3-7 illustrates the frequency response for an inverse Chebyshev reference analog band-stop filter of the 4th order.  
To design inverse Chebyshev reference analog pototype filter, it is necessary to know the filter order.
Inverse Chebyshev analog filter is defined via expression:
where: Ω is the frequency;
N is the filter order;
ε is the parameter of maximum oscillation in the passband frequency response; and
TN is the Chebyshev polynomial.

The Chebyshev polynomial TN(Ω) can be obtained from recursive relation:
If the filter order is familiar in advance, neither these recursive relations nor expression for the square of frequency response are necessary. The design process starts from the values of poles of a 1st order inverse Chebyshev reference analog filter.

The poles of the transfer function of inverse Chebyshev analog filter are considered reciprocal poles of the transfer function of a 1st order Chebyshev analog filter.

The poles of a 1st order Chebyshev analog filter are expressed as:
where:
si is the i-th pole of the transfer function of analog prototype filter (complex value);
σi is the real pole; and
Ωi is the imaginary pole.
where:
N is the filter order; and
i=1, 2, ..., N.

The poles of the transfer function of inverse Chebyshev analog filter are found via expression:
where:
si is the pole of the transfer function of a 1st order Chebyshev analog filter; and
s2i is the pole of the transfer function of inverse Chebyshev analog filter.

Transfer function is expressed as:
The coefficient k in numerator can be only an odd number. Table 3-3-1 provides a few examples of values of k.

N	MIN	MAX	VALUES
5	1	5	1, 3, 5
6	1	5	1, 3, 5
7	1	7	1, 3, 5, 7
8	1	7	1, 3, 5, 7

Table 3-3-1. coefficient k in the transfer function numerator

The values Ωk are found via expression:
The value H0 is found via expression:
For N=5, the transfer function is:    
As seen from Figure 3-3-8 and expression for Ωk, the zeros of the transfer function are always complex-conjugated values, which is not the case with the poles of the transfer function.